## Optimization (Motivation)

Briefly, we consider the problem when a tangent line to a curve has zero slope. That is, the derivative of the curve is zero at some point.

Such a point is called a “critical point” and tells us about extreme values.

Example. Consider the function $f(x)=x^{2}-2x+1$. We plot this out:

Now, we can consider the tangent line at, say, $x=1$. We draw this tangent line in red:

Observe every point on the line $(x,f(x))$ has the property that $f(1)\leq f(x)$. In other words, $f(1)$ is a “Global Minimum” (it’s less than every other $f(x)$ if $x\not=1$).

How can we detect such “extrema” (i.e., “maxima” and “minima”)? Well, the tangent line has zero slope. Equivalently: $f'(x_{0})=0$ when the point $(x_{0}, f(x_{0}))$ is an extreme point.

Sometimes these “extrema” points are local. For example, plotting $f(x)=x^{-1}\cos(3\pi x)$ when $0.1\leq x\leq1.1$:

So how do we detect extrema?

Well, we take our function $g(x)$, take its derivative, then find points $x_{0}$ such that $g'(x_{0})=0$.

This is the general routine, but lets leave the motivating post with a few questions…

Problem: as we can see with the function $x^{-1}\cos(3\pi x)$, some extrema are local while others are global. How do we
(a) Determine if an extrema is a maximum or minimum?
(b) Determine if the extrema is local or global?